Geometric Langlands Correspondence: Further Directions

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The Langlands Program is a mathematical framework that unifies questions in many different areas of mathematics, especially number theory and representation theory, in a web of deep and as yet only partially understood connections between number theory, representation theory, geometry, and mathematical physics. The classical Langlands program has been studied for more than fifty years, and has found significant applications to solving classical Diophantine equations, such as the solution of Fermat's last theorem. The geometric Langlands program, which is relatively new, is under rapid development as it is also connected with other subjects such as geometry and physics, from where one can draw intuition. In the geometric Langlands program, the number fields in the classical Langlands program are replaced by curves and their function fields, and the web of connections in number theory is replaced by concisely formulated equivalences, or correspondences. This project will extend and develop geometric Langlands correspondences in different settings and provide research training opportunities for graduate students.In more detail, one project will establish the Fundamental Local Equivalence, which is an equivalence between the Whittaker category of the affine Grassmannian for a reductive group and the Kazhdan-Lusztig category for its Langlands dual, as factorization categories. This equivalence could be considered as the starting point for the local geometric Langlands program. The project will establish the required equivalence by equating both sides to a combinatorial object that is directly expressible in terms of the root data and the quantum parameter (the so-called factorization algebra Omega). Another project will directly relate the geometric and classical Langlands theories (in the case of function fields) by realizing the space of automorphic functions as the categorical trace of the Frobenius on the category of automorphic sheaves with nilpotent singular support. In the process of doing so, one naturally re-derives V. Lafforgue's decomposition of the space of automorphic functions over the course moduli space of Langlands parameters via shtukas.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.